Heegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformization

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1 Invent. math. 131, 453±491 (1998) Heegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformization Massimo Bertolini 1, Henri Darmon 2 1 UniversitaÁ di Pavia, Dipartimento di Matematica. I.-Universita degli studi di Pavia, Strada Nuova, 65, I Pavia, Italy ( massimodragon.ian.pv.cnr.it) 2 McGill University, Mathematics Department. 805 Sherbrooke St. West, Montreal, PQ Canada H3A-2K6 ( darmon@math.mcgill.ca) Oblatum 24-V-1996 & 17-IV-1997 Contents Introduction Quaternion algebras, upper half planes, and trees The p-adic L-function Generalities on Mumford curves Shimura curves Heegner points Computing the p-adic Abel-Jacobi map Proof of the main results References Introduction Let E=Q be a modular elliptic curve of conductor N, and let K be an imaginary quadratic eld. Rankin's method gives the analytic continuation and functional equation for the Hasse-Weil L-function L E=K; s. When the sign of this functional equation is 1, a Heegner point a K is de ned on E K using a modular curve or a Shimura curve parametrization of E. In the case where all the primes dividing N are split in K, the Heegner point comes from a modular curve parametrization, and the formula of Gross- Zagier [GZ] relates its Ne ron-tate canonical height to the rst derivative of L E=K; s at s ˆ 1. Perrin-Riou [PR] later established a p-adic analogue of the Gross-Zagier formula, expressing the p-adic height of a K in terms of a derivative of the 2-variable p-adic L-function attached to E=K. At around the same time, Mazur, Tate and Teitelbaum [MTT] formulated a p-adic Birch and Swinnerton-Dyer conjecture for the p-adic L-function of E associated to the

2 454 M. Bertolini, H. Darmon cyclotomic Z p -extension of Q, and discovered that this L-function acquires an extra zero when p is a prime of split multiplicative reduction for E. The article [BD1] proposed analogues of the Mazur-Tate-Teitelbaum conjectures for the p-adic L-function of E associated to the anticyclotomic Z p -extension of K.Ina signi cant special case, the conjectures of [BD1] predict a p-adic analytic construction of the Heegner point a K from the rst derivative of the anticyclotomic p-adic L-function. (Cf. conjecture 5.8 of [BD1].) The present work supplies a proof of this conjecture. We state a simple case of our main result; a more general version is given in Sect. 7. Assume from now on that N is relatively prime to disc K, that E is semistable at all the primes which divide N and are inert in K=Q, and that there is such a prime, say p. Let O K be the ring of integers of K, and let u K :ˆ 1 2 #O K : (Thus, u p K ˆ 1 unless K ˆ Q i or Q 3 :) Note that the curve E=K p has split multiplicative reduction, and thus is equipped with the Tate p-adic analytic uniformization U Tate : K p! E K p ; whose kernel is the cyclic subgroup of Kp generated by the Tate period q 2 pz p. Let H be the Hilbert class eld of K, and let H 1 be the compositum of all the ring class elds of K of conductor a power of p. Write G 1 :ˆ Gal H 1 =H ; ~G 1 :ˆ Gal H 1 =K ; D :ˆ Gal H=K : By class eld theory, the group G 1 is canonically isomorphic to Kp =Q p O K, which can also be identi ed with a subgroup of the group Kp;1 of elements of Kp of norm 1, by sending z to z z u K, where z denotes the complex conjugate of z in Kp. A construction of [BD1], Sect. 2.7 and 5.3, based on ideas of Gross [Gr], and recalled in Sect. 2, gives an element L p E=K in the completed integral group ring Z ~G 1 Š which interpolates the special values of the classical L-function of E=K twisted by complex characters of ~G 1. We will show (Sect. 2) that L p E=K belongs to the augmentation ideal ~I of Z ~G 1 ŠŠ. Let L 0 p E=K denote the image of L p E=K in ~I=~I 2 ˆ ~G 1. The reader should view L 0 p E=K 2 ~G 1 as the rst derivative of L p E=K evaluated at the central point. One shows that the element L 0 p E=K actually belongs to G 1 ~G 1, so that it can (and will) be viewed as an element of Kp of norm 1. Using the theory of Jacquet-Langlands, and the assumption that E is modular, we will de ne a surjective map g f : J! ~E, where ~E is an elliptic curve isogenous to E over Q, and J is the Jacobian of a certain Shimura curve X. The precise de nitions of X, J, g f and ~E are given at the end of Sect. 4. At the cost of possibly replacing E with an isogenous curve, we assume from now on in the introduction that E ˆ ~E. (This will imply that E is the ``strong Weil curve'' for the Shimura curve parametrization).

3 Heegner points, p-adic L-functions 455 A special case of our main result is: Theorem A. The local point U Tate L 0 p E=K in E K p is a global point in E K. When L 0 p E=K is non-trivial, theorem A gives a construction of a rational point on E K from the rst derivative of the anticyclotomic p-adic L-function of E=K, in much the same way that the derivative at s ˆ 0 of the Dedekind zeta-function of a real quadratic eld leads to a solution of Pell's equation. A similar kind of phenomenon was discovered by Rubin [Ru] for elliptic curves with complex multiplication, with the exponential map on the formal group of E playing the role of the Tate parametrization. See also a recent result of Ulmer [U] for the universal elliptic curve over the function eld of modular curves over nite elds. We now state theorem A more precisely. In Sect. 5, a Heegner point a K 2 E K is de ned as the image by g f of certain divisors supported on CM points of X. Let a K be the complex conjugate of a K. Theorem B. Let w ˆ 1(resp. w ˆ 1) if E=Q p multiplicative reduction. Then has split (resp. non-split) U Tate L 0 p E=K ˆ a K wa K : Theorem B, which relates the Heegner point a K to the rst derivative of a p- adic L-function, can be viewed as an analogue in the p-adic setting of the theorem of Gross-Zagier, and also of the p-adic formula of Perrin-Riou [PR]. Unlike these results, it does not involve heights of Heegner points, and gives instead a p-adic analytic construction of a Heegner point. Observe that G 1 is isomorphic to Z p Z= p 1 Z, so that its torsion subgroup is of order p 1. Choosing an anticyclotomic logarithm k mapping G 1 onto Z p determines a map from Z G 1 Š to the formal power series ring Z p T Š. Let L p E=K be the image of L p E=K in Z p T Š, and L 0 p E=K the derivative of L p E=K with respect to T evaluated at T ˆ 0. Since U Tate is injective on Kp;1, theorem B implies: Corollary C. The derivative L 0 p E=K is non-zero if and only if the point a K wa K is of in nite order. Corollary C gives a criterion in terms of the rst derivative of a p-adic L-function for a Heegner point coming from a Shimura curve parametrization to be of in nite order. Work in progress of Keating and Kudla suggests that a similar criterion (involving the Heegner point a K itself) can be formulated in terms of the rst derivative of the classical L-function, in the spirit of the Gross-Zagier formula. The work of Kolyvagin [Ko] shows that if a K is of in nite order, then E K has rank 1 and III E=K is nite. By combining this with corollary C, one obtains

4 456 M. Bertolini, H. Darmon Corollary D. If L 0 p E=K is non-zero, then E K has rank 1 and III E=K is nite. The formula of theorem B is a consequence of the more general result given in Sect. 7, which relates certain Heegner divisors on jacobians of Shimura curves to derivatives of p-adic L-functions. The main ingredients in the proof of this theorem are (1) a construction, based on ideas of Gross, of the anticyclotomic p-adic L-function of E=K, (2) the explicit construction of [GVdP] of the p-adic Abel-Jacobi map for Mumford curves, and (3) the Cerednik-Drinfeld theory of p-adic uniformization of Shimura curves. 1 Quaternion algebras, upper half planes, and trees De nite quaternion algebras Let N be a product of an odd number of distinct primes, and let B be the (unique, up to isomorphism) de nite quaternion algebra of discriminant N. Fix a maximal order R B. (There are only nitely many such maximal orders, up to conjugation by B ). For each prime `, we choose certain local orders in B` :ˆ B Q`, as follows. 1. If ` is any prime which does not divide N, then B` is isomorphic to the algebra of 2 2 matrices M 2 Q` over Q`. Any maximal order of B` is isomorphic to M 2 Z`, and all maximal orders are conjugate by B`.We x the maximal order R` :ˆ R Z`: 2. If ` is a prime dividing N, then B` is the (unique, up to isomorphism) quaternion division ring over Q`. We let R` :ˆ R Z`; as before. The valuation on Z` extends uniquely to R`, and the residue eld of R` is isomorphic to F`2, the nite eld with `2 elements. We x an orientation of R`, i.e., an algebra homomorphism o ` : R`! F ` 2 : Note that there are two possible choices of orientation for R`. 3. For each prime ` which does not divide N, and each integer n 1, we also choose certain oriented Eichler orders of level `n. These are Eichler orders R n ` of level `n contained in R`, together with an orientation of level `n, i.e., an algebra homomorphism

5 Heegner points, p-adic L-functions 457 o ` : R n `! Z =`nz: We will sometimes write R` for the oriented Eichler order R 1 ` of level `. For each integer M ˆ `ni Qi i which is prime to N, let R M be the (oriented) Eichler order of level M in R associated to our choice of local Eichler orders: 0 1 R M :ˆ B \ Ỳ R` R n i A: 6jM We view R M as endowed with the various local orientations o ` and o ` for the primes ` which divide MN, and call such a structure an orientation on R M. We will usually view R M as an oriented Eichler order, in what follows. Let ^Z ˆ Q` Z` be the pro nite completion of Z, and let ^B :ˆ B ^Z ˆ Y be the adelization of B. Likewise, if R 0 is any order in B (not necessarily maximal), let ^R 0 :ˆ R 0 ^Z. The multiplicative group ^B acts (on the left) on the set of all oriented Eichler orders of a given level M by the rule b R 0 :ˆ B \ b ^R 0 b 1 ; b 2 ^B ; R 0 B: (Note that b R 0 inherits a natural orientation from the one on R 0.) This action of ^B is transitive, and the stabilizer of the oriented order R M is precisely ^R M. Hence the choice of R M determines a description of the set of all oriented Eichler orders of level M, as the coset space ^R M n ^B. Likewise, the conjugacy classes of oriented Eichler orders of level M are in bijection with the double coset space ^R M n ^B =B : Let N be an integer which is prime to N, and let p be a prime which does not divide N N. We set ` N ˆ N N p: Let C be the group of elements in R N 1 p Š of reduced norm 1. Of course, the de nition of C depends on our choice of local orders, but: Lemma 1.1. The group C depends on the choice of the R` and R n `, only up to conjugation in B. Proof. This follows directly from strong approximation ([Vi], p. 61). `i B` `i

6 458 M. Bertolini, H. Darmon The p-adic upper half plane attached to B Fix an unrami ed quadratic extension K p of Q p. De ne the p-adic upper half plane (attached to the quaternion algebra B) as follows: H p :ˆ Hom K p ; B p : Remark. The group GL 2 Q p acts naturally on P 1 K p by MoÈ bius transformations, and the choice of an isomorphism g : B p! M 2 Q p determines an identi cation of H p with P 1 K p P 1 Q p. This identi cation sends w 2 H p to one of the two xed points for the action of gw Kp on P1 K p. More precisely, it sends w to the unique xed point P 2 P 1 K p such that the induced action of Kp on the tangent line T P P 1 K p ˆ K p is via the character z 7! z z. More generally, a choice of an embedding B p! M 2 K p determines an isomorphism of H p with a domain X in P 1 K p. In the literature, the p-adic upper half plane is usually de ned to be P 1 C p P 1 Q p ˆC p Q p, where C p is the completion of (an) algebraic closure of Q p. From this point of view, it might be more appropriate to think of H p as the K p -rational points of the p-adic upper half plane. But in this work, the role of the complex numbers in the p-adic context is always played, not by C p, but simply (and more naively) by the quadratic extension K p. We will try as much as possible to work with the more ``canonical'' de nition of the upper half plane, which does not depend on a choice of embedding of B p into M 2 K p. The upper half plane H p is endowed with the following natural structures. 1. The group B p acts naturally on the left on H p, by conjugation. This induces a natural action of the discrete group C on H p. 2. An involution w 7! w, de ned by the formula: w z :ˆ w z ; where z 7! z is the complex conjugation on K p. The Bruhat-Tits tree attached to B Let T be the Bruhat-Tits tree of B p =Q p. The vertices of T correspond to maximal orders in B p, and two vertices are joined by an edge if the intersection of the corresponding orders is an Eichler order of level p. An edge of T is a set of two adjacent vertices on T, and an oriented edge of T is an ordered pair of adjacent vertices of T. We denote the set of edges (resp. oriented edges) of T by E T (resp.! E T ).

7 Heegner points, p-adic L-functions 459 The edges of T correspond to Eichler orders of level p, and the oriented edges are in bijection with the oriented Eichler orders of level p. Since T is a tree, there is a distance function de ned on the vertices of T in a natural way. We de ne the distance between a vertex v and an edge e to be the distance between v and the furthest vertex of e. The group B p acts on T via the rule b R 0 :ˆ br 0 b 1 ; b 2 B p ; R 0 2 T: This action preserves the distance on T. In particular, the group C acts on T by isometries. Fix a base vertex v 0 of T. A vertex is said to be even (resp. odd) if its distance from v 0 is even (resp. odd). This notion determines an orientation on the edges of T, by requiring that an edge always go from the even vertex to the odd vertex. The action of the group B p does not preserve the orientation, but the subgroup of elements of norm 1 (or, more generally, of elements whose norm has even p-adic valuation) sends odd vertices to odd vertices, and even ones to even ones. In particular, the group C preserves the orientation we have de ned on T. The reduction map Let O p be the ring of integers of K p. Given w 2 H p, the image w O p is contained in a unique maximal order R w of B p. In this way, any w 2 H p determines a vertex R w of T. We call the map w 7! R w the reduction map from H p to T, and denote it r : H p! T: For an alternate description of the reduction map r, note that the map w from K p to B p determines an action of Kp on the tree T. The vertex r w is the unique vertex which is xed under this action. The lattice M Let G :ˆ T=C be the quotient graph. Since the action of C is orientation preserving, the graph G inherits an orientation from T. Let E G be the set of (unordered) edges of G, and let V G be its set of vertices. Write Z E G Š and Z V G Š for the modules of formal Z-linear combinations edges and vertices of G, respectively. There is a natural boundary (compatible with our : Z E G Š! Z V G Š

8 460 M. Bertolini, H. Darmon which sends an edge fa; bg to a b, with the convention that a is the odd vertex and b is the even vertex in fa; bg. There is also a coboundary map de ned : Z V G Š! Z E G v ˆ X e; ~v2e where the sum is taken over the images in E G of the p 1 edges of T containing an arbitrary lift ~v of v to T. The sign in the formula is 1 if v is odd, and 1 ifvis even. Recall the canonical pairings de ned by Gross [Gr] on Z E G Š and on Z V G Š. Ifeis an edge (resp. v is a vertex) de ne w e (resp. w v )tobethe order of the stabilizer for the action of C of (some) lift of e (resp. v) tot. Then he i ; e j iˆw ei d ij ; hhv i ; v j ii ˆ w vi d ij : Extend these pairings by linearity to the modules Z E G Š and Z V G Š. Lemma 1.2. The are adjoint with respect to the pairings h ; i and hh ; ii, i.e., he;@ viˆhh@ e;vii: Proof. By direct computation. De ne the module M as the quotient M :ˆ Z E G : Given two vertices a and b of T, they are joined by a unique path, which may be viewed as an element of Z E G Š in the natural way. Note that because of our convention for orienting T, if a and b are even vertices (say) joined by 4 consecutive edges e 1, e 2, e 3 and e 4, then the path from a to b is the formal sum path a; b ˆe 1 e 2 e 3 e 4 2ZEG Š: Note that we have the following properties of the path function: path a; b ˆ path b; a ; path a; b path b; c ˆpath a; c :

9 Heegner points, p-adic L-functions 461 Also, if a and b are C-equivalent, then path a; b belongs to H 1 G; Z Z E G Š. Proposition 1.3. The map from M to Hom C; Z which sends m 2 M to the function is injective and has nite cokernel. c 7! hpath v 0 ; cv 0 ; mi Proof. The pairing h ; i gives an injective map with nite cokernel But M! Hom ; Z : ˆH 1 G;Z : Let C ab denote the abelianization of C. Then the map of C ab to H 1 G; Z which sends c to path v 0 ; cv 0 is an isomorphism modulo torsion (cf. [Se]). The proposition follows. Relation of M with double cosets We now give a description of M in terms of double cosets which was used in [BD1], Sect More precisely, let J N p; N h i ˆ Z ^R N p n ^B =B be the lattice de ned in [BD1], Sect (By previous remarks, the module J N p;n is identi ed with the free Z-module ZR 1 ZR t generated by the conjugacy classes of oriented Eichler orders of level N p in the quaternion algebra B.) Likewise, let J N ;N ˆ Z ^R N n ^B =B Š: In [BD1], Sect. 1.7, we de ned two natural degeneracy maps J N ;N! J N p ; N ;

10 462 M. Bertolini, H. Darmon and a module J p new N p;n to be the quotient of J N p;n by the image of J N ; N J N ;N under these degeneracy maps. Proposition 1.4. The choice of the oriented Eichler order R N p determines an isomorphism between M and J p new N p;n. The proof of proposition 1.4 uses the following lemma: Lemma 1.5. There exists an element c 2 R N 1 p Š whose reduced norm is an odd power of p. Proof. Let F be an auxiliary imaginary quadratic eld of prime discriminant such that all primes dividing N are split in F and all primes dividing N are inert in F. Such an F exists, by Dirichlet's theorem on primes in arithmetic progressions. By genus theory, F has odd class number, and hence its ring of integers O F contains an element a of norm p k, with k odd. Fix an embedding of O F in the Eichler order R N, and let c be the image of a in R N 1 p Š. Proof of proposition 1.4. Recall that R p B p denotes our xed local Eichler order of level p. By strong approximation, we have ^R N p n ^B =B ˆ R p Q p nb p =R N 1 : p The group R p Q p is the stabilizer of an ordered edge of T. Hence R p Q p nb p is identi ed with the set! E T of ordered edges on T, and the double coset space R p Q p nb p =R N 1 p Š is identi ed with the set of ordered edges! E G on the quotient graph G :ˆ T=R N 1. p But the map which sends fx; yg 2E G to x; y 2E! G if x is even, and to y; x if x is odd, is a bijection between E G and! E G. For, if fx; yg and fx 0 ; y 0 g have the same image in! E G, then there is an element of R N 1 p Š which sends the odd vertex in fx; yg to the odd vertex in fx 0 ; y 0 g and the even vertex in fx; yg to the even vertex in fx 0 ; y 0 g. This element is necessarily in C, since it sends an odd vertex to an odd vertex. Hence the edges fx; yg and fx 0 ; y 0 g are C-equivalent, and our map is one-one. To check surjectivity, let c be! the element of R N 1 p Š given by lemma 1.5. Then the element x; y of E G is the image of fx; yg if x is even and y is odd, and is the image of fcx; cyg if x is odd and y is even. To sum up, we have shown that the choice of the Eichler order R N p determines a canonical bijection between J N p;n and Z E G Š. Likewise, one shows that the Eichler order R N determines a canonical bijection between J N ;N and the set of vertices V G, and between J N ;N J N ;N and Z V G Š. (The resulting map from Z V G Š Z V G Š to Z V G Š sends a pair v; w to v w, where where v and w are lifts of v and w to vertices of G, which are even and odd respectively.) Finally, from the de nition of the degeneracy maps given in [BD1] one checks that the following diagram commutes up to sign:

11 Heegner points, p-adic L-functions 463 J N ;N J N ;N! Z V G Š # ; J N p;n! Z E G Š where the horizontal maps are the identi cations we have just established, and the left vertical arrow is the di erence of the two degeneracy maps. (Which is only well-de ned up to sign). From this, it follows that M ˆ Z E G is identi ed with the module of [BD1]. J p new N p;n ˆ J N p;n =image J N ;N J N ;N Hecke operators The lattice M is equipped with a natural Hecke action, coming from its description in terms of double cosets. (Cf. [BD1], Sect. 1.5.) Let T be the Hecke algebra acting on M. Recall that N ˆ N N p: The following is a consequence of the Eichler trace formula, and is a manifestation of the Jacquet-Langlands correspondence between automorphic forms on GL 2 and quaternion algebras. Proposition 1.6. If / : T! C is any algebra homomorphism, and a n ˆ / T n for all n with gcd n; N p ˆ1 ;then the a n are the Fourier coe cients of a normalized eigenform of weight 2 for C 0 N. Conversely, every normalized eigenform of weight 2 on C 0 N which is new at p and at the primes dividing N corresponds in this way to a character /. Given a normalized eigenform f on X 0 N, denote by O f the order generated by the Fourier coe cients of f and by K f the fraction eld of O f. Assuming that f is new at p and at N, let p f 2 T K f be the idempotent associated to f by proposition 1.6. Let n f 2 O f be such that g f :ˆ n f p f belongs to T O f. Let M f M O f be the sublattice on which T acts via the character associated to f. The endomorphism g f induces a map, still denoted g f by an abuse of notation, g f : M! M f : In particular, if f has integer Fourier coe cients, then M f is isomorphic to Z. Fixing such an isomorphism (i.e., choosing a generator of M f ), we obtain a map which is well-de ned up to sign. g f : M! Z;

12 464 M. Bertolini, H. Darmon 2 The p-adic L-function We recall the notations and assumptions of the introduction: E is a modular elliptic curve of conductor N, associated to an eigenform f on C 0 N ; K is a quadratic imaginary eld of discriminant D relatively prime to N. Furthermore: 1. the curve E has good or multiplicative reduction at all primes which are inert in K=Q; 2. there is at least one prime, p, which is inert in K and for which E has multiplicative reduction; 3. the sign in the functional equation for L E=K; s is 1. Write N ˆ N N p; where N, resp. N is divisible only by primes which are split, resp. inert in K. Note that by our assumptions, N is square-free and not divisible by p. Lemma 2.1. Under our assumptions, N is a product of an odd number of primes. Proof. By page 71 of [GZ], the sign in the functional equation of the complex L-function L E=K; s is 1 #f`jn pg 1. The result follows. Let c be an integer prime to N. We modify slightly the notations of the introduction, letting H denote now the ring class eld of K of conductor c, and H n the ring class eld of conductor cp n. We write H 1 ˆ S H n, and set G n :ˆ Gal H n =H ; ~G n :ˆ Gal H n =K ; G 1 :ˆ Gal H 1 =H ; ~G 1 :ˆ Gal H 1 =K ; D :ˆ Gal H=K : (Thus, the situation considered in the introduction corresponds to the special case where c ˆ 1.) There is an exact sequence of Galois groups 0! G 1! ~G 1! D! 0; and, by class eld theory, G 1 is canonically isomorphic to Kp =Q p O k. The completed integral group rings Z G 1 ŠŠ and Z ~G 1 ŠŠ are de ned as the inverse limits of the integral group rings Z G n Š and Z ~G n Š under the natural projection maps. We set M G n Š :ˆ M Z G n Š; M G 1 Š :ˆ lim M G n ŠˆMZ G 1 Š; n

13 Heegner points, p-adic L-functions 465 and likewise for G n and G 1 replaced by ~G n and ~G 1. The groups G 1 and ~G 1 act naturally on M G 1 Š and M ~G 1 ŠŠ by multiplication on the right. In this section, we review the construction of a p-adic L-function L p M=K, in a form adapted to the calculations we will perform later. A slightly modi ed version of this construction is given in Sect 2.7 of [BD1]. It is based on results of Gross [Gr] on special values of the complex L-functions attached to E=K, and on their generalization by Daghigh [Dag]. Let ZZ X f :ˆ 4p 2 jf s j 2 ds ^ ids H 1 =C be the complex period associated to the cusp form f. Write d for the discriminant of the order O of conductor c, u for one half the order of the group of units of O and n f for the integer de ned at the end of section 1 by the relation g f ˆ n f p f. Theorem 2.2. There is an element L p M=K 2M ~G 1 Š, well-de ned up to right multiplication by ~G 1, with the property that jv g f L p M=K j 2 ˆ L f =K; v; 1 X f p d nf u k 2 ; for all nite order complex characters v of ~G 1 associated to T as in proposition 1.6. and all modular forms f Proof. See [Gr], [Dag] and [BD1], Sect 2.7. Corollary 2.3. Setting L p E=K :ˆ g f L p M=K 2 Z ~G 1 Š; where f is the modular form associated to E, one has jv L p E=K j 2 L E=K; v; 1 p ˆ d nf u k 2 ; X f for all nite order characters v of ~G 1. Remark. One sees that the interpolation property of corollary 2.3 determines L p E=K uniquely, up to right multiplication by elements in ~G 1, if it exists. The existence amounts to a statement of rationality and integrality for the special values L E=K; v; 1. The construction of L p M=K (and hence, of L p E=K ) is based on the notion of Gross points of conductor c and cp n. Gross points of conductor c Recall that O is the order of conductor c in the maximal order O K, where we assume that c is prime to N. We equip O with an orientation of level N N, i.e., for each `njjn, an algebra homomorphism

14 466 M. Bertolini, H. Darmon o ` : O! Z=`nZ; and for each `jn, an algebra homomorphism o ` : O! F ` 2 : An embedding n : O! R n of O into an oriented Eichler order R n of level dividing N is called an oriented embedding if it respects the orientations on O and on R n, i.e., if the diagrams O! n R n O! n R n # o ` # o ` # o ` # o ` Z=`nZ ˆ Z=`nZ F`2 ˆ F`2 commute, for all ` which divide N N. The embedding n is called optimal if it does not extend to an embedding of any larger order into R n. The group B acts naturally on the set of oriented optimal embeddings of conductor c, by conjugation: b R n ; n :ˆ br n b 1 ; bnb 1 : De nition 2.4 A Gross point of conductor c and level N N is a pair R n ; n where R n is an oriented Eichler order of level N in B, and n is an oriented optimal embedding of O into R n, taken modulo conjugation by B. We denote by Gr c the set of all Gross points of conductor c and level N N. Given n 2 Hom K; B, we denote by ^n 2 Hom ^K; ^B the natural extension of scalars. The group D ˆ Pic O ˆ^O n^k nk acts on the Gross points, by the rule r R n ; n :ˆ ^n r R n ;n : Lemma 2.5. The group D acts simply transitively on the Gross points of conductor c. Proof. See [Gr], Sect. 3. One says that R n ; n is in normal form if

15 Heegner points, p-adic L-functions 467 R n Z` ˆ R` for all `6 jnp; R n Z` ˆ R n ` as oriented Eichler orders, for all `njjn ; R n Z` ˆ R` as oriented orders, for all `jn : (Note in particular that we have imposed no condition on R n Z p in this de nition.) Choose representatives R 1 ; w 1, R 2 ; w 2,..., R h ;w h for the Gross points of conductor c, written in normal form. (This can always be done, by strong approximation.) Note that 1 R i ˆ R 1 as oriented Eichler orders, p p and that the orders R i are completely determined by the local order R i Z p. Let v 1 ;...;v h be the vertices on T associated to the maximal orders R 1 Z p,...,r h Z p. The vertex v i is equal to r w i, i.e., it is the image of w i (viewed as a point on H p in the natural way) by the reduction map to T. Gross points of conductor cp n Let n 1, and let O n denote the order of K of conductor cp n. De nition 2.6. A Gross point of conductor cp n and level N is a pair R n ; n where R n is an oriented Eichler order of level N p in B, and n is an oriented optimal embedding of O n into R n, taken modulo conjugation by B. To make De nition 2.6 complete, we need to clarify what we mean by an orientation at p of the optimal embedding n. (For the primes which divide N N, the meaning is exactly the same as before.) The oriented Eichler order R n Z p corresponds to an ordered edge on T, whose source and target correspond to maximal orders R 1 and R 2 respectively. We require that n still be an optimal embedding of O n into R 2. (It then necessarily extends to an optimal embedding of O n 1 into R 1.) We let Gr cp n be the set of Gross points of level cp n, and we set Gr cp 1 :ˆ [1 nˆ1 Gr cp n : The group ~G n ˆ ^O n n ^K =K acts on Gr cp n by the rule r R n ; n :ˆ ^n r R n ;n :

16 468 M. Bertolini, H. Darmon Lemma 2.7. The group ~G n acts simply transitively on Gr cp n. Proof. See [Gr], Sect. 3. In particular, the group ~G 1 acts transitively on Gr cp 1. As before, we say that a Gross point R n ; n of conductor cp n is in normal form if R n Z` ˆ R` for all `6 jnp; R n Z` ˆ R n ` as oriented Eichler orders, for all `njjn ; R n Z` ˆ R` as oriented orders, for all `jn : Recall the representatives R 1 ; w 1 ;...; R h ;w h for the Gross points of conductor c that were chosen in the previous paragraph. Lemma 2.8. Every point in Gr cp 1 is equivalent to an element in normal form, and can be written as R 0 ; w i, where w i 2fw 1 ;...;w h g, and R 0 Z p is an oriented Eichler order of level p. A point in Gr cp 1 described by a pair R 0 ; w i is of level cp n, where n is the distance between the edge associated to R 0 on T, and the vertex associated to R i. Proof. The rst statement follows from strong approximation, and the second from a direct calculation. By Lemma 2.8, the set Gr cp 1 can be described by the system of representatives E T fw;...;w h g: The action of G 1 ˆ Kp =Q p on Gr cp1 in this description is simply r R 0 ; w i :ˆ ^w i r R 0 ;w i : Construction of L p M=K Choose one of the representatives of Gr c, say, v 1 ; w 1. Choose an end of T originating from v 1, i.e., a sequence e 1 ; e 2 ;...;e n ;...of consecutive edges originating from v 1. By Lemma 2.8, the Gross points e n ; w 1 are a sequence of Gross points of conductor cp n. Consider the formal expression 1 X n r e n ; w 1 r 1 r2 G ~ n and let L p;n M=K denote its natural image in M ~ G n Š. Lemma 2.9. The elements L p;n M=K (n 1 are compatible under the natural projection maps M ~G n 1 Š!M ~G n Š.

17 Heegner points, p-adic L-functions 469 Proof. This follows directly from the de niton of the action of ~G n on Gr cp n given above, and from the de nition of the coboundary They yield that the formal expression Norm Kn 1 =K n e n 1 ; w 1 e n ;w 1 is in the image of the coboundary and hence is zero in M. The lemma follows. Lemma 2.9 implies that we can de ne an element L p M=K 2 M ~G 1 Š by taking inverse limit of the L p;n M=K via the projections M G n 1 Š!M G n Š. The element L p M=K satis es the conclusions of Theorem 2.2. It should be thought of as a p-adic L-function (or rather, the square root of a p-adic L-function) over K, associated to modular forms for T. Iff is any such modular form, then the element g f L p M=K is equal to the element h N ;N de ned in [BD1], Sect. 5.3 (in the special case when f has rational coe cients). Note that L p M=K depends on the choice of the initial point v 1 ; w 1, and on the end e 1 ;...;e n ;... of T originating from v 1, but only up to multiplication (on the right) by an element of ~G 1. Recall the augmentation ideal I of Z ~G 1 Š described in the introduction. More generally, let I D be the kernel of the augmentation map Z ~G 1 ŠŠ! Z DŠ. Lemma L p M=K belongs to M I. In fact, L p M=K belongs to M I D. Proof. Since D acts simply transitively on v 1 ; w 1 ;...; v h ;w h, let r i be the element such that r i v 1 ˆ v i : Let I D denote, by abuse of notation, the image of I D in Z ~G n Š. Note that we have the canonical isomorphisms Z ~G 1 Š=I D ˆ Z ~G n =ID ˆ Z DŠ: By the compatibility lemma 2.9, the image of L p M=K in M ~G 1 ŠŠ=I D is equal to the image of L p;1 M=K in M G 1 Š=I D ˆ M DŠ, which is equal to: X h iˆ1! X e v i 2e r 1 i : But each of the terms in the inner sum belongs to the image and hence is 0 in M. Thus, L p M=K belongs to M I D, and also to M I, since I D I.

18 470 M. Bertolini, H. Darmon Remark. If v is any character of D and f is any modular form attached to T, then the functional equation of L f =K; v; s has sign 1, and hence L f =K; v; 1 ˆ0 for all such characters. The interpolation formula of Theorem 2.2 implies then that L p M=K belongs to I D. The point of the proof of Lemma 2.10 is that the construction of L p M=K also implies this directly, without using the relation with L-function values. Let L 0 p M=K 2 M I=I2 ˆ M ~G 1 and L 0 p M=H 2 M I D=ID 2 ˆ M D ŠG1ˆ M DŠ K p;1 be the natural images of the element L p M=K. Since L p M=K is wellde ned up to right multiplication by ~G 1, the element L 0 p M=K is canonical, and does not depend on the choice of v 1 ; w 1 or on the choice of the end of T originating from v 1. The element L 0 p M=H is well de ned, up to right multiplication by an element of D. We now give an explicit description of L 0 p M=K and L0 p M=H in Hom C; Kp;1 which will be used in the calculations of Sects. 6 and 7. Let w be any point in H p, corresponding to a local embedding of K p into B p. The embedding w gives rise to an action of Kp =Q p on the tree T by multiplication on the right, xing the vertex v 0 :ˆ r w. Choose a sequence of ends e 1 ;...;e n ;... originating from v 0, and let L 0 p;n w ˆ 1 n X r2g n w r e n r 1 be the element of M G n (here we denote by e n the element in M associated to the edge e n ). The elements L 0 p;n w are compatible under the obvious projection maps M G n 1! M G n, and hence the element L 0 p w 2M G 1 can be de ned as the inverse limit of the L 0 p;n w under the natural projections. By proposition 1.3, we may view L 0 p w as an element of Hom C; Kp;1, given by L 0 p w d ˆlim n D E path v 0 ; dv 0 ; L 0 p;n w 2 G 1 ˆ Kp;1 ; 8d 2 C: In this notation, we have L 0 p M=K ˆ Xh iˆ1 L 0 p w i ;

19 Heegner points, p-adic L-functions 471 L 0 p M=H ˆ X r2d L 0 p wr 1 r 1 : 3 Generalities on Mumford curves Following [Jo-Li], we call a smooth complete curve X over K p an admissible curve over K p if it admits a model X over the ring of integers O p of K p, such that: (i) the scheme X is proper and at over O p ; (ii) the irreducible components of the special ber X p are rational and de ned over O p = p 'F p 2, and the singularities of X p are ordinary double points de ned over O p = p ; (iii) if x 2 X p is a singular point, then the completion ^O X;x of the local ring O X;x is O-isomorphic to the completion of the local ring O X ; Y Š= XY p m for a positive integer m. Let C be a nitely generated subgroup of PGL 2 K p, acting on P 1 C p by MoÈ bius transformations. A point z 2 P 1 C p is said to be a limit point for the action of C if it is of the form z ˆ lim g n z 0 for a sequence of distinct elements g n of C. Let I P 1 C p denote its set of limit points and let X p ˆ P 1 K p I. The group C is said to act discontinuously, ortobea discontinuous group, ifx p 6ˆ ;. A fundamental result of Mumford, extended by Kurihara, establishes a 1-1 correspondence between conjugacy classes of discontinuous groups and admissible curves. Theorem 3.1. Given an admissible curve X over K p, there exists a discontinuous group C PGL 2 K p, unique up to conjugation, such that X K p is isomorphic to X p =C. Conversely, any such quotient is an admissible curve over K p. Proof. See [Mu] and [Ku]. If D ˆ P 1 P r Q 1 Q r 2Div X p is a divisor of degree zero on X p, de ne the theta function z cp 1 z cp r h z; D ˆY z cq 1 z cq r ; c2c with the convention that z 1ˆ1. Let C ab :ˆ C= C; CŠ be the abelianization of C, and let C :ˆ C ab = C ab tor be its maximal torsion-free quotient. Lemma 3.2. There exists / D 2 Hom C; K p such that h dz; D ˆ/ D d h z;d,for all d in C. Furthermore, the map / D factors through C, so that / D can be viewed as an element of Hom C; K p.

20 472 M. Bertolini, H. Darmon Proof. See [GVdP], p. 47, (2.3.1), and ch. VIII, prop. (2.3). Let U AJ : Div X p!hom C; Kp be the map which associates to the degree zero divisor D the automorphy factor / D. The reader should think of this map as a p-adic Abel-Jacobi map. Given d 2 C, the number / z dz b does not depend on the choice of z 2 X p, and depends only on the image of a and b in C. Hence it gives rise to a well-de ned pairing ; : C C! K p : Lemma 3.3. The pairing ; is bilinear, symmetric, and positive de nite (i.e., ord p ; is positive de nite). Hence, the induced map j : C! Hom C; Kp is injective and has discrete image. Proof. See [GVdP], VI.2. and VIII.3. Given a divisor D of degree zero on X K p ˆX p =C, let ~D denote an arbitrary lift to a degree zero divisor on X p. Let K :ˆ j C. The automorphy factor / D ~ depends on the choice of ~D, but its image in Hom C; Kp =K depends only on D. Thus U AJ induces a map Div X K p! Hom C; Kp =K, which we also call U AJ by abuse of notation. Proposition 3.4. The map Div X K p! Hom C; Kp =K de ned above is trivial on the group of principal divisors, and induces an identi cation of the K p -rational points of the jacobian J of X over K p with Hom C; Kp =K. Proof. See [GVdP], VI.2. and VIII.4. To sum up, we have: Corollary 3.5. The diagram commutes. Div 0 X p U AJ! Hom C; Kp # # Div 0 X K p J K p U AJ!

21 Heegner points, p-adic L-functions Shimura curves Let B be the inde nite quaternion algebra of discriminant N p, and let R be an (oriented) maximal order in B (which is unique up to conjugation). Likewise, for each M prime to N p, choose an oriented Eichler order R M of level M contained in R. Let X be the Shimura curve associated to the Eichler order R N,asin [BD1], sec I Moduli description of X The curve X =Q is a moduli space for abelian surfaces with quaternionic multiplication and N -level structure. More precisely, the curve X =Q coarsely represents the functor F Q which associates to every scheme S over Q the set of isomorphism classes of triples A; i; C, where 1. A is an abelian scheme over S of relative dimension 2; 2. i : R! End S A is an inclusion de ning an action of R on A; 3. C is an N -level structure, i.e., a subgroup scheme of A which is locally isomorphic to Z=N Z and is stable and locally cyclic under the action of R N. See [BC], ch. III and [Rob] for more details. Remarks. 1. The datum of the level N structure is equivalent to the data, for each `n jn, of a subgroup C` which is locally isomorphic to Z=`nZ and is locally cyclic for the action of R N. 2. For each ` dividing N p, let I R` be the maximal ideal of R`. The subgroup scheme A I of points in A killed by I is a free R`=I ' F`2-module of rank one, and the orientation o ` : R`! F ` 2 allows us to view A I canonically as a one-dimensional F`2-vector space. II Complex analytic description of X Let B 1 :ˆ B R ' M 2 R : De ne the complex upper half plane associated to B to be H 1 :ˆ Hom C; B 1 : Note that a choice of isomorphism g : B 1! M 2 R determines an isomorphism of H 1 with the union C R of the ``usual'' complex upper half plane fz 2 C : Imz > 0g

22 474 M. Bertolini, H. Darmon with the complex lower half plane, by sending w 2 Hom C; B 1 to the unique xed point P of gw C such that the induced action of C on the complex tangent line T P C R ˆCis by the character z 7! z z. Let C 1 ˆ R N be the group of invertible elements in R N (i.e., having reduced norm equal to 1). This group acts naturally on H 1 via the action of B 1 by conjugation. Proposition 4.1. The Shimura curve X over C is isomorphic to the quotient of the complex upper half plane H 1 attached to B 1 by the action of C 1, i.e., X C ˆH 1 =C 1 : Proof. See [BC], ch. III, and [Rob]. In particular, an abelian surface A over C with quaternionic multiplications by R and level N structure determines a point w 2 H 1 ˆ Hom C; B 1 which is well-de ned modulo the natural action of C 1. We will now give a description of the assignment A 7! w. Although not used in the sequel, this somewhat non-standard description of the complex uniformization is included to motivate the description of the p-adic uniformization of X which follows from the work of Cerednik and Drinfeld. The complex upper half plane as a moduli space. We rst give a ``moduli'' description of the complex upper half plane H 1 :ˆ Hom C; B 1 as classifying complex vector spaces with quaternionic action and a certain ``rigidi cation''. De nition 4.2. A quaternionic space (attached to B 1 is a two-dimensional complex vector space V equipped with a (left) action of B 1, i.e., an injective homomorphism i : B 1! End C V. Let V R be the 4-dimensional real vector space underlying V. Lemma 4.3. The algebra End B1 V R is isomorphic (non-canonically) to B 1. Proof. The natural map B 1 End B1 V R!End R V R 'M 4 R is an isomorphism, and hence End B1 V R is abstractly isomorphic to the algebra B 1. De nition 4.4. A rigidi cation of the quaternionic space V is an isomorphism q : B 1! End B1 V R : A pair V ; q consisting of a quaternionic space V and a rigidi cation q is called a rigidi ed quaternionic space.

23 Heegner points, p-adic L-functions 475 There is a natural notion of isomorphism between rigidi ed quaternionic spaces. Proposition 4.5. There is a canonical bijection between H 1 and the set of isomorphism classes of rigidi ed quaternionic spaces. Proof. Given w 2 H 1 ˆ Hom C; B 1, we de ne a rigidi ed quaternionic space as follows. Let V ˆ B 1, viewed as a two-dimensional complex vector space by the rule kv :ˆ vw k ; v 2 V ; k 2 C: The left multiplication by B 1 on V endows V with the structure of quaternionic space. The right multiplication of B 1 on V is then used to de ne the rigidi cation B 1! End B1 V R. Conversely, given a rigidi ed quaternionic space V ; q, one recovers the point w in H 1 by letting w k be q 1 m k, where m k is the endomorphism in End B1 V R induced by multiplication by the complex number k. One checks that these two assignments are bijections between H 1 and the set of isomorphism classes of rigidi ed quaternionic spaces, and that they are inverses of each other. We now describe the isomorphism X C ˆH 1 =C 1 given in proposition 4.1. Let A be an abelian surface over C with quaternionic multiplication by R and level N structure. Then the Lie algebra V ˆ Lie A is a quaternionic space in a natural way. (The quaternionic action of B 1 is induced by the action of R on the tangent space, by extension of scalars from Z to R.) Moreover, V is equipped with an R-stable sublattice K which is the kernel of the exponential map V! A. Lemma The endomorphism ring End R K is isomorphic (non-canonically) to R. 2. The set of endomorphisms in End R K which preserve the level N - structure on K is isomorphic (non-canonically) to the Eichler order R N. Proof. The natural map B End R K Q!End Q K Q 'M 4 Q is an isomorphism, and hence End R K Qis abstractly isomorphic to the quaternion algebra B. Furthermore, the natural map End R K!End Z K

24 476 M. Bertolini, H. Darmon has torsion-free cokernel, and hence End R K is a maximal order in B. Likewise, one sees that the subalgebra of End R K preserving the level N structure (viewed as a submodule of 1 N K=K) is an Eichler order of level N. Fix an isomorphism having the following properties. q 0 : R! End R K ; 1. For each `n jn, q 0 R N Z` preserves the subgroup C` (viewed as a subgroup of 1`n K=K). By the remark 1 above, R N operates on C` via a homomorphism R N!Z=`nZ. In addition, we require that this homomorphism be equal to the orientation o `. 2. For all `jn p, the algebra R` acts on 1` K=K, and stabilizes the subspace V corresponding to A I (where I is the maximal ideal of R`). By the remark 2 above, V is equipped with a canonical F`2-vector space structure, and q 0 R` acts F`2-linearly on it. We require that the resulting homomorphism R`! F ` 2 be equal to the orientation o `. With these conventions, the homomorphism q 0 is well-de ned, up to conjugation by elements in C 1. Let q : B 1! End B1 V R be the map induced from q 0 by extension of scalars from Z to R. The pair V ; q is a rigidi ed quaternionic space, which depends only on the isomorphism class of A, up to the action of C 1 on q by conjugation. The pair V ; q thus gives a well-de ned point on H 1 =C 1 associated to A. It is a worthwhile exercise for the reader to check that this complex analytic description of the moduli of abelian varieties with quaternionic multiplications corresponds to the usual description of the moduli space of elliptic curves as H 1 =SL 2 Z, in the case where the quaternion algebra B is M 2 Q. III p-adic analytic description of X The fundamental theorem of Cerednik and Drinfeld states that X is an admissible curve over Q p and gives an explicit description of the discrete subgroup attached to X by theorem 3.1. More precisely, let B, R, and C R N 1 p Š be as in section 1. (So that B is the de nite quaternion algebra obtained from B by the Cerednik ``interchange of invariants'' at p.) Then we have: Theorem 4.7 (Cerednik-Drinfeld). The set of K p -rational points of the Shimura curve X is isomorphic to the quotient of the p-adic upper half plane H p attached to B by the natural action of C, i.e., X K p ˆH p =C:

25 Heegner points, p-adic L-functions 477 Under this identi cation, the involution w 7! w of H p corresponds to the involution sw p of X K p, where s is the complex conjugation in Gal K p =Q p, and w p is the Atkin-Lehner involution of X at p. Proof. See [C], [Dr] and [BC]. In particular, an abelian surface A over K p with quaternionic multiplications by R and level N structure determines a point w 2 H p ˆ Hom K p ; B p which is well-de ned modulo the natural action of C. We will now give a precise description of the assignment A 7! w. Crucial to this description is Drinfeld's theorem that the p-adic upper half plane H p parametrizes isomorphism classes of certain formal groups with a quaternionic action, and a suitable ``rigidi cation''. The p-adic upper half plane as a moduli space. We review Drinfeld's moduli interpretation of the (K p -rational points of the) p-adic upper half plane H p. Roughly speaking, H p classi es formal groups of dimension 2 and height 4 over O p, equipped with an action of our xed local order R p and with a ``rigidi cation'' of their reduction modulo p. In order to make this precise, we begin with a few de nitions. Let as usual k be O p = p ' F p 2. De nition 4.8. A 2-dimensional commutative formal group V over O p is a formal R p -module (for brevity, a FR-module) if it has height 4 and there is an embedding i : R p! End V : The FR-modules play the role of the quaternionic spaces of the previous section. Let V be the formal group over k deduced from V by extension of scalars from O p to k. It is equipped with the natural action of R p given by reduction of endomorphisms. Let End 0 V :ˆ End V Q p be the algebra of quasi-endomorphisms of V, and let End 0 B p V be the subalgebra of quasiendomorphisms which commute with the action of B p. Lemma The algebra End 0 V is isomorphic (non-canonically) to M 2 B p. 2. The algebra End 0 B p V is isomorphic (non-canonically) to the matrix algebra B p over Q p. Proof. The formal group V is isogenous to the formal group of a product of two supersingular elliptic curves in characteristic p. Part 1 follows. Part 2 can then be seen by noting that the natural map B p End 0 B p V!End 0 V 'M 2 B p

26 478 M. Bertolini, H. Darmon is an isomorphism, so that End 0 B p V is abstractly isomorphic to the matrix algebra B p. Denote by B p;u the subgroup of elements of B p whose reduced norm is a p- adic unit. De nition Arigidi cation of the FR-module V is an isomorphism q : B p! End 0 B p V ; subject to the condition of being ``positively oriented at p'', i.e., that the two maximal orders R p and q 1 End Rp V of B p are conjugated by an element of B p;u. 2. A pair V ; q consisting of an FR-module V and a rigidi cation q is called a rigidi ed FR-module. 3. Two rigidi ed modules V ; q and V 0 ; q 0 are said to be isomorphic if there is an isomorphism / : V! V 0 of formal groups over O p, such that the induced isomorphism satis es the relation / q ˆ q 0. / : End 0 B p V!End 0 B p V 0 Remark. In [Dr] and [BC], a rigidi cation of a FR-module V is de ned to be a quasi-isogeny of height zero from a xed FR-module U to the reduction V modulo p of V. This de nition is equivalent to the one we have given, once one has xed an isomorphism between B p and End 0 B p U. The de nition given above is in a sense ``base-point free''. Recall that B p;u acts (on the left) on H p via the natural action of B p on H p by conjugation. Note that B p;u acts on the left on (the isomorphism classes of) rigidi ed FR-modules, by b V ; q :ˆ V;q b for b in B p;u ; where q b x is equal to q b 1 xb for x in B p. Theorem 4.11 (Drinfeld). 1. The p-adic upper half plane H p is a moduli space for the isomorphism classes of rigidi ed FR-modules over O p. In particular, there is a bijective map W : f V ; q : V ; q a rigidified FR moduleg= isomorphisms Hom K p; B p :! 2. The map W is B p;u -equivariant.

27 Heegner points, p-adic L-functions 479 Proof. See [Dr] and [BC], chapters I and II. For part 2, see in particular [BC], ch. II, Sect. 9. Corollary All FR-modules have formal multiplication by O p. Proof. If V is a FR-module, equip V with a rigidi cation q. By theorem 4.11, the pair V ; q determines a point P V ;q of the p-adic upper half plane H p. Note that the stabilizer of P V ;q for the action of B p;u is isomorphic to O p. The claim now follows from part 2 of theorem Remark. As we will explain in the next paragraph, if V is an FR-module, there exists an abelian surface A over O p with quaternionic multiplication by R, whose formal group ^A (with the induced action of R p ) is isomorphic to V. Of course, quite often one has End R A 'Z, even though End R V contains O p by corollary In fact, combining Drinfeld's theory with the theory of complex multiplication shows the existence of an uncountable number of such abelian surfaces such that (i) End R A ˆZ; (ii) End Rp ^A 'O p.(a similar phenomenon for elliptic curves has been observed by Lubin and Tate [LT].) We give a description of the bijection W, which follows directly from Drinfeld's theorem. By lemma 4.12, identify End Rp V with O p. Let w : K p! B p be the map induced by the composition O p! End 0 B p V!B p ; where the rst map is given by the reduction modulo p of endomorphisms, and the second map is just q 1. Then W V ; q ˆw. We now use Drinfeld's theorem to describe the p-adic uniformization of the K p -rational points of the Shimura curve X, i.e., the isomorphism X K p ˆH p =C: The curve X has a model X over Z p. Given a point in X K p, we may extend it to a point in X O p. In other words, given a pair A; i; C, where A is an abelian surface over K p with quaternionic action by i, and C is a level N - structure, we may extend it to a similar pair A; i; C of objects over O p.we write A;i; C for the reduction modulo p of A; i; C. A p-quasi endomorphism of A is an element in End A Z 1 pš. The algebra of all p-quasi endomorphisms is denoted by End p A. Likewise, we denote by End p R A the algebra of p-quasi-endomorphisms which commute with the action of R. Let B p1 be the quaternion algebra over Q rami ed at p and 1, and let R p1 be a maximal order of B p1.